Method of Kinematic Ranging

ABSTRACT

A method of kinematic ranging for finding the range R of a jammer moving on a trajectory involves measuring the bearing of the jammer and the rate of change thereof using an airborne detector radar at a first position ( 24 ), causing the airborne detector radar to carry out a manoeuvre such that is it displaced in the horizontal plane by a displacement having orthogonal components Δx, Δy, and measuring the bearing of the jammer at a second position subsequent to the manoeuvre. By making an appropriate choice for the components Δx, Δy, the range R may be found with a desired relative range accuracy, and the error in R may be minimized.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 to European application 14 000 178.5, filed Jan. 17, 2014, the entire disclosure of which is herein expressly incorporated by reference.

BACKGROUND AND SUMMARY OF THE INVENTION

Exemplary embodiments of the invention relate to kinematic ranging techniques.

Kinematic ranging per se is a known electronic counter-counter-measure technique, in which the range of a moving airborne jammer may be estimated by use of a radar unit without the need for the unit to actively interrogate the jammer, i.e. the radar unit may work in a passive mode to detect and process the jammer signals. In the specific examples of the invention described below, the radar unit is an airborne detecting radar unit. Kinematic ranging is described for example in “Kinematic Ranging for IRST”, SPIE Vol 1950, Acquisition, Tracking and Pointing VII (1993).

Referring to FIG. 1, in a known kinematic ranging technique, an airborne jammer moves along a trajectory 18 from a first position 10 to a second position 12 in a time interval Δt=t₂−t₁, during which an airborne detecting radar unit carries out a manoeuvre along a path 20 from a position 14 to a position 16, corresponding to a displacement 22 defined by displacement components Δx, Δy in the x and y directions respectively. The change Δα in azimuth bearing angle of the jammer with respect to the airborne detecting radar unit between the times t₁ and t₂ is given by

${{\Delta \; \alpha} = {\alpha_{1} + {{\frac{\alpha}{t} \cdot \Delta}\; t} - \alpha_{2}}};$

where α₁ and α₂ are the bearings in azimuth of the jammer at times t₁ and t₂ with respect to positions 14 and 16 respectively. The angle Δβ between the straight line connecting positions 14 and 10 and the displacement 22 is given by

$\begin{matrix} {{\Delta \; \beta} = {\frac{\pi}{2} - {\arctan \; \frac{\Delta \; y}{\Delta \; x}} - \alpha_{1} - {{\frac{\alpha}{t} \cdot \Delta}\; {t.}}}} & \left( {{Equation}\mspace{14mu} 1} \right) \end{matrix}$

Δβ is the angle in the azimuthal plane between the straight line joining the position of the airborne detector radar at time t₁ and the position of the jammer at t₂ and the straight line corresponding to the displacement of the airborne detector radar during the interval Δt. Using the sine rule, the range R of the jammer from the airborne detecting radar at time t₂ is given by

$\begin{matrix} {\frac{R}{\sin \; \Delta \; \beta} = {\frac{\sqrt{{\Delta \; x^{2}} + {\Delta \; y^{2}}}}{\sin \; \Delta \; \alpha}.}} & \left( {{Equation}\mspace{14mu} 2} \right) \end{matrix}$

The range R of the jammer from the second position 16 of the airborne detecting radar may be therefore be calculated by measuring the bearings α₁, α₂ in azimuth of the jammer at the times t₁, t₂ with respect to the airborne detecting radar unit, the rate of change dα/dt of the bearing in azimuth of the jammer with respect to the position 14, the components Δx, Δy of the displacement 22 corresponding to the manoeuvre 20 of the airborne detecting radar, and the time interval Δt.

A disadvantage of the known method of kinematic ranging is that when the airborne detecting radar unit carries out a general manoeuvre 20, corresponding to a general displacement such as 22, the final result for the range R typically involves a significant level of uncertainty as a result of measurement errors associated with the values of α₁, α₂, dα/dt, Δt, Δx and Δy.

The present invention provides a method of kinematic ranging comprising the steps of:

-   -   (i) measuring the bearing α₁ in azimuth of a jammer at a time t₁         with respect to a first position using an airborne radar         detector located at the first position;     -   (ii) measuring the rate of change dα/dt of the bearing in         azimuth of the jammer with respect to the first position using         the airborne radar detector located at the first position;     -   (iii) causing the airborne radar detector to carry out a         manoeuvre such that it is displaced to a second position by a         horizontal displacement d having orthogonal components Δx, Δy in         a time Δt=t₂−t₁ and measuring the bearing α₂ in azimuth of the         jammer at time t₂ with respect to the second position using the         airborne radar detector located at the second position, where         d=√{square root over (Δx²+Δy²)};     -   (iv) calculating the difference Δα in the bearing in azimuth of         the jammer between the second and first positions;         characterised in that the components Δx, Δy are calculated by         the steps of:     -   (a) choosing a desired relative range accuracy σ_(R)/R for the         method;     -   (b) obtaining an estimated range R_(est) of the jammer from the         second position at time t₂;     -   (c) finding d on the basis of the relative range accuracy, the         estimated range R_(est), the variance σ_(Δα) ² in Δα and the         variance σ_(d) ² in d according to

$\sigma_{R} = {\frac{R}{d}\sqrt{{\left( {R^{2} - d^{2}} \right) \cdot \sigma_{\Delta \; \alpha}^{2}} + \sigma_{d}^{2}}}$

-   -   (d) calculating the components Δx, Δy of the displacement d         according to

Δx=cos(α₁+{dot over (α)}·Δt)·d

Δy=−sin(α₁+{dot over (α)}·Δt)·d

and also characterised in that the range R of the jammer from the second position is calculated according to

$R = {\frac{d}{\sin \; \Delta \; \alpha}.}$

The invention provides the advantage that the range R may be measured with a desired relative range accuracy by causing the airborne radar detector to carry out the manoeuvre d having the calculated displacement components Δx, Δy in the horizontal plane.

In order to provide the possibility of a further increase in the accuracy of the determined range R, preferably the method further comprises the steps of

-   -   (i) evaluating the angle Δβ₁ between the straight line joining         the first position of the airborne detector radar and the         position of the jammer at time t₂, and the straight line defined         by the components Δx, Δy;     -   (ii) evaluating the angle Δβ₂ between the straight line joining         the first position of the airborne detector radar and the         position of the jammer at time t₂, and the straight line defined         by the components −Δx, Δy;     -   (iii) if |Δβ₁−π/2|≦|Δβ₂−π/2| then choosing the manoeuvre of the         airborne detector radar such that its displacement has         components Δx, Δy and if |Δβ₁−π/2|>|Δβ₂−π/2| then choosing the         manoeuvre of the airborne detector radar such that its         displacement has components −Δx, Δy.

In some embodiments, the method may be applied to the simultaneous determination of the range of two jammers, in which case the method preferably further includes the steps of:

-   -   (i) evaluating the angle Δβ₁₁ between the straight line joining         the first position of the airborne detector radar and the         position of the first jammer at time t₂, and the straight line         defined by the components Δx, Δy;     -   (ii) evaluating the angle Δβ₁₂ between the straight line joining         the first position of the airborne detector radar and the         position of the second jammer at time t₂, and the straight line         defined by the components Δx, Δy;     -   (iii) evaluating the angle Δβ₂₁ between the straight line         joining the first position of the airborne detector radar and         the position of the first jammer at time t₂, and the straight         line defined by the components −Δx, Δy;     -   (iv) evaluating the angle Δβ₂₂ between the straight line joining         the first position of the airborne detector radar and the         position of the second jammer at time t₂, and the straight line         defined by the components −Δx, Δy;     -   (v) if |Δβ₁₁+Δβ₁₂−π|≦|Δβ₂₁+Δβ₂₂−π| then choosing the manoeuvre         of the airborne detector radar such that its displacement has         components Δx, Δy and if     -   |Δβ₁₁+Δβ₁₂−π|>|Δβ₂₁+Δβ₂₂−π| then choosing the manoeuvre of the         airborne detector radar such that its displacement has         components −Δx, Δy.

This ensure the greatest overall accuracy in the determined ranges of the two jammers.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

Embodiments of the invention are described below by way of example only, and with reference to the accompanying drawings in which:

FIG. 1 illustrates a known kinematic ranging technique;

FIG. 2 shows the relationship between the range R of a jammer and the displacement d of a manoeuvre of an airborne radar detector executed in a method of the invention in the case of a relative range accuracy of less than 20%;

FIG. 3 illustrates the measurement of the range R of a jammer with minimum uncertainty;

FIG. 4 illustrates a non-optimised manoeuvre of an airborne detector radar for estimating the range of multiple jammers; and

FIG. 5 illustrates an alternative manoeuvre of an airborne detector radar for estimating the ranges of multiple jammers with improved accuracy.

DETAILED DESCRIPTION

Using Gauss' law in conjunction with Equation 2, the variance in the range R of the jammer is given by

$\begin{matrix} {\sigma_{R}^{2} = {{\left( \frac{\partial R}{\partial\left( {\Delta \; \beta} \right)} \right)^{2}\sigma_{\Delta \; \beta}^{2}} + {\left( \frac{\partial R}{\partial\left( {\Delta \; a} \right)} \right)^{2}\sigma_{\Delta \; a}^{2}} + {\left( \frac{\partial R}{\partial d} \right)^{2}\sigma_{d}^{2}}}} & \left( {{Equation}\mspace{14mu} 3} \right) \end{matrix}$

where σ² _(Δβ), σ² _(Δα), and σ² _(d) are the variances in the angles Δβ, Δα and in the displacement d respectively. Inserting the derivatives of Equation 2 into Equation 3 gives an expression for the relative range accuracy σ_(R)/R:

${\frac{\sigma_{R}^{2}}{R^{2}} = {{{\cot^{2}\left( {\Delta \; \beta} \right)}\sigma_{\Delta \; \beta}^{2}} + \frac{\sigma_{\Delta \; a}^{2}}{\tan^{2}\left( {\Delta \; a} \right)} + \frac{\sigma_{d}^{2}}{d^{2}}}},$

which is minimised when

cot²(Δβ)=0

and Δα and d are as large as possible. cot² (Δβ)=0 if Δβ=π/2; combining this with Equation 1 gives

${\arctan \; \frac{\Delta \; y}{\Delta \; x}} = {{- \alpha_{1}} - {{\overset{.}{\alpha} \cdot \Delta}\; t}}$

or equivalently

Δy=−tan(α₁+{dot over (α)}·Δt)·Δx

For Δβ=π/2, Equations 2 and 3 can be written as

$\begin{matrix} {{R = \frac{d}{\sin \left( {\Delta \; \alpha} \right)}}{and}{\sigma_{R} = {\frac{R}{d}\sqrt{{\left( {R^{2} - d^{2}} \right) \cdot \sigma_{\Delta \; \alpha}^{2}} + \sigma_{d}^{2}}}}} & \left( {{Equation}\mspace{14mu} 4} \right) \end{matrix}$

FIG. 2 shows a plane through the function of Equation 4 at the value σ_(R)/R=0.2 assuming accuracies σ_(d), σ_(Δα) in the measurement of d and Δα of 700 m and 1° respectively. If an initial value of the range R of 70 km is estimated, from FIG. 2 this corresponds to a displacement d of ˜7000 m. The values α₁, dα/dt are measured when the airborne detector radar is at position 14, as described above with reference to FIG. 1. The time Δt taken for the airborne detector radar to carry out the manoeuvre 20 in order to cover the displacement d is d/v+dt where v is the velocity of the radar detector and dt is a tolerance value.

${\Delta \; t} = {{\frac{7000\mspace{14mu} m}{250\mspace{14mu} m\text{/}s} + {5\mspace{14mu} s}} = {33\mspace{14mu} s}}$

Taking d=7000 m, dt=5 s and v=250 m/s, and assuming that α₁=−20° and {dot over (α)}=0.2°/s, the components Δx, Δy of the displacement d are

Δx=cos(α₁+{dot over (α)}·Δt)·d=cos(−20°+6.6°)·d≈6810 m

Δy=−sin(α₁+{dot over (α)}·Δt)·d=−sin(−20°+6.6°)·d≈1620 m

If the airborne detector radar carries out the manoeuvre 20 using these components of the displacement d, then the error in the measurement of the range R has a relative range accuracy of 20% or better. FIG. 3 shows the displacement 32 of length d, the angle Δβ being 90° in this case. In FIG. 3, the jammer moves from a position 20 to a position 22 along a trajectory 28 in a time interval Δt, during which the airborne detector radar moves from a position 24 to a position 26, these positions defining the displacement d.

FIG. 4 illustrates how the method may be applied to find the range of multiple jammers by increasing the number of manoeuvre s carried out by a detecting airborne radar. In this case, n jammers are located and assessed at time t₁ and have bearings in azimuth of α₁₁ . . . α_(n1). The airborne detector radar carries out a manoeuvre on a path 40 from point 34 to the end 36 which define a displacement 42 of length d=(Δx²+Δy²)^(1/2). At time t₂ at the position 36, the measured bearings in azimuth of the jammers are α₁₂ . . . α_(n2). The differences in bearing Δα_(n) and the angle Δβ_(n) are

${\Delta \; \alpha_{n}} = {\alpha_{n\; 1} + {{\frac{\alpha_{n}}{t} \cdot \Delta}\; t} - \alpha_{n\; 2}}$ ${\Delta \; \beta_{n}} = {\frac{\pi}{2} - {\arctan \; \frac{\Delta \; y}{\Delta \; x}} - \alpha_{n\; 1} - {{\frac{\alpha_{n\;}}{t} \cdot \Delta}\; t}}$

and the range R_(n) of the nth jammer is given by

$R_{n} = {\frac{{\sin \left( {\Delta \; \beta_{n}} \right)}\sqrt{{\Delta \; x^{2}} + {\Delta \; y^{2}}}}{\sin \; \Delta \; \alpha_{n}}.}$

In FIG. 4, the value Δβ₁ is approximately 60° which leads to acceptable results, however Δβ_(n) may become close to zero, so that the relative range accuracy for the measurement of R_(n) approaches a pole. To avoid this, the manoeuvre of the airborne detector radar may be carried out so that the displacement Δx is negative, as shown in FIG. 5; in this case a displacement 52 between positions 44 and 46 is achieved by a manoeuvre 50. Using this manoeuvre 50, Δβ_(n) is approximately 90°, so that the error in the range R is minimised, and Δβ₁ is approximately 60°, which results in the accuracy of the measurement of R₁ also being acceptable. In this example, although the manoeuvre is optimised for jammer 1, the angle velocity dα/dt of jammer 1 increases Δβ₁ without lowering Δβ_(n); this is because both jammers have the same direction of angle velocity dα/dt.

The foregoing disclosure has been set forth merely to illustrate the invention and is not intended to be limiting. Since modifications of the disclosed embodiments incorporating the spirit and substance of the invention may occur to persons skilled in the art, the invention should be construed to include everything within the scope of the appended claims and equivalents thereof. 

What is claimed is:
 1. A method of kinematic ranging comprising the steps of: (i) measuring a bearing al in azimuth of a jammer at a time t₁ with respect to a first position using an airborne radar detector located at a first position; (ii) measuring a rate of change dα/dt of the bearing in azimuth of the jammer with respect to the first position using the airborne radar detector located at the first position; (iii) causing the airborne radar detector to carry out a manoeuvre such that it is displaced to a second position by a horizontal displacement d having orthogonal components Δx, Δy in a time Δt=t₂−t₁ and measuring a bearing α₂ in azimuth of the jammer at time t₂ with respect to the second position using the airborne radar detector located at the second position, where d=√{square root over (Δx²+Δy²)}; (iv) calculating a difference Δα in the bearing in azimuth of the jammer between the second and first positions; wherein components Δx, Δy are calculated by the steps of (a) choosing a desired relative range accuracy σ_(R)/R for the method; (b) obtaining an estimated range R_(est) of the jammer from the second position at time t₂; (c) finding d based on the relative range accuracy, the estimated range R_(est), the variance σ_(Δα) ² in Δα and the variance σ_(d) ² in d according to $\sigma_{R} = {\frac{R}{d}\sqrt{{\left( {R^{2} - d^{2}} \right) \cdot \sigma_{\Delta \; \alpha}^{2}} + \sigma_{d}^{2}}}$ (d) calculating the components Δx, Δy of the displacement d according to Δx=cos(α₁+{dot over (α)}·Δt)·d Δy=−sin(α₁+{dot over (α)}·Δt)·d wherein a range R of the jammer from the second position is calculated according to $R = {\frac{d}{\sin \; \Delta \; \alpha}.}$
 2. The method of claim 1, further comprising the steps of: (i) evaluating an angle Δβ₁ between a straight line joining the first position of the airborne detector radar and the position of the jammer at time t₂, and a straight line defined by the components Δx, Δy; (ii) evaluating an angle Δβ₂ between the straight line joining the first position of the airborne detector radar and the position of the jammer at time t₂, and a straight line defined by the components −Δx, Δy; (iii) if |Δβ₁−π/2|≦|Δβ₂−π/2| then choosing the manoeuvre of the airborne detector radar such that the airborne detector radar's displacement has components Δx, Δy and if |Δβ₁−π/2|>|Δβ₂−π/2| then choosing the manoeuvre of the airborne detector radar such that the airborne detector radar's displacement has components −Δx, Δy.
 3. The method according to claim 1, wherein a range of a second jammer is also determined, and wherein the method comprises the steps of: (i) evaluating an angle Δβ₁₁ between a straight line joining the first position of the airborne detector radar and the position of the first jammer at time t₂, and a straight line defined by the components Δx, Δy; (ii) evaluating an angle Δβ₁₂ between a straight line joining the first position of the airborne detector radar and the position of the second jammer at time t₂, and the straight line defined by the components Δx, Δy; (iii) evaluating an angle Δβ₂₁ between the straight line joining the first position of the airborne detector radar and the position of the first jammer at time t₂, and a straight line defined by the components −Δx, Δy; (iv) evaluating an angle Δβ₂₂ between the straight line joining the first position of the airborne detector radar and the position of the second jammer at time t₂, and the straight line defined by the components −x, −y; (v) if |Δβ₁₁+Δβ₁₂−π|≦|Δβ₂₁+Δβ₂₂−π| then choosing the manoeuvre of the airborne detector radar such that the airborne detector radar's displacement has components Δx, Δy and if |Δβ₁₁+Δβ₁₂−π|>|Δβ₂₁+Δβ₂₂−π| then choosing the v of the airborne detector radar such that the airborne detector radar's displacement has components −Δx, Δy. 